To expand \((a + 1)^3\) using Pascal's Triangle, we first need to identify the coefficients for the third row of Pascal's Triangle, which corresponds to \(n=3\).
The third row of Pascal's Triangle is:
- Row 0: \(1\)
- Row 1: \(1, 1\)
- Row 2: \(1, 2, 1\)
- Row 3: \(1, 3, 3, 1\)
The coefficients for the expansion of \((a + 1)^3\) are \(1, 3, 3, 1\).
Using these coefficients, the expansion is:
\[ (a + 1)^3 = 1 \cdot a^3 + 3 \cdot a^2 \cdot 1 + 3 \cdot a \cdot 1^2 + 1 \cdot 1^3 \]
This simplifies to:
\[ = a^3 + 3a^2 + 3a + 1 \]
So the expansion of \((a + 1)^3\) is:
\[ \boxed{a^3 + 3a^2 + 3a + 1} \]
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